It is a difficult technical problem to obtain an explicit expression for \(\mathfrak {C}_{\alpha (n)}(h)\)  in terms of the AdS covariant derivative. In particular, the minimal uplift of (REF ) is not gauge invariant or transverse. To restore these properties, (REF ) must be supplemented with curvature dependent terms, as in the low spin cases (). However, for generic spin, previous attempts to do this were unsuccessful [1]}, [2]}. In order to derive the Cotton tensors in AdS\(_3\) , it is advantageous to recast the problem into the framework of homogeneous polynomials (or generating functions). The basics of this formalism are discussed in appendix .
